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Properties

Label 2-273-91.74-c1-0-3
Degree $2$
Conductor $273$
Sign $0.731 - 0.681i$
Analytic cond. $2.17991$
Root an. cond. $1.47645$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.831·2-s + (0.5 + 0.866i)3-s − 1.30·4-s + (−1.30 − 2.26i)5-s + (−0.415 − 0.719i)6-s + (1.78 + 1.95i)7-s + 2.75·8-s + (−0.499 + 0.866i)9-s + (1.08 + 1.88i)10-s + (0.924 + 1.60i)11-s + (−0.654 − 1.13i)12-s + (2.74 + 2.33i)13-s + (−1.48 − 1.62i)14-s + (1.30 − 2.26i)15-s + 0.331·16-s + 6.83·17-s + ⋯
L(s)  = 1  − 0.587·2-s + (0.288 + 0.499i)3-s − 0.654·4-s + (−0.585 − 1.01i)5-s + (−0.169 − 0.293i)6-s + (0.673 + 0.738i)7-s + 0.972·8-s + (−0.166 + 0.288i)9-s + (0.343 + 0.595i)10-s + (0.278 + 0.482i)11-s + (−0.188 − 0.327i)12-s + (0.761 + 0.648i)13-s + (−0.396 − 0.434i)14-s + (0.337 − 0.585i)15-s + 0.0828·16-s + 1.65·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.731 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273\)    =    \(3 \cdot 7 \cdot 13\)
Sign: $0.731 - 0.681i$
Analytic conductor: \(2.17991\)
Root analytic conductor: \(1.47645\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{273} (256, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 273,\ (\ :1/2),\ 0.731 - 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.828697 + 0.325998i\)
\(L(\frac12)\) \(\approx\) \(0.828697 + 0.325998i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (-1.78 - 1.95i)T \)
13 \( 1 + (-2.74 - 2.33i)T \)
good2 \( 1 + 0.831T + 2T^{2} \)
5 \( 1 + (1.30 + 2.26i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.924 - 1.60i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 6.83T + 17T^{2} \)
19 \( 1 + (2.53 - 4.39i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.27T + 23T^{2} \)
29 \( 1 + (-0.724 + 1.25i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.09 - 5.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.87T + 37T^{2} \)
41 \( 1 + (-4.41 + 7.64i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.109 - 0.189i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.624 - 1.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.33 + 2.32i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 12.0T + 59T^{2} \)
61 \( 1 + (-4.36 + 7.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.91 + 11.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.78 - 3.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.26 - 5.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.08 - 5.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.67T + 83T^{2} \)
89 \( 1 + 15.1T + 89T^{2} \)
97 \( 1 + (-6.08 - 10.5i)T + (-48.5 + 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.13765396938422315948782307399, −10.93801212619375801233885584050, −9.825692145016378008068612101296, −9.039410791129737983491409035771, −8.348544853803201447372651120576, −7.73924248527826972616332859791, −5.71107270475927664882825014889, −4.66941691218197822335558178707, −3.84591363846646054857197669544, −1.48577486305277133139762548770, 1.01038734614467746215122824393, 3.19016661966284561660177921283, 4.28382769991409376523461792515, 5.92142337250900525162389811332, 7.36341528432067934783024829182, 7.82267398268925229595116823733, 8.710883437532803834860790578013, 9.943631270564270523035448379693, 10.84744218970497641128800073550, 11.46649228659099017807377722898

Graph of the $Z$-function along the critical line